Most instruments tend to pitch sharp or flat depending on the partial. I don't recall any music teachers giving advice that specific positions should routinely get adjustments, but instead that notes in a particular partial should be adjusted. For example, F above middle C should be flattened when played in 1st position to compensate for 6th partial tending sharp. Or the G in 2nd position above that F needing to be pulled in a bit to compensate for 7th partial tending flat.

Manufacturers have different philosophies around this as well. I have a vintage mid-1960s King 3b whose partials line up differently and require different adjustment from my modern XO 1634... and both of those horns are extremely similar .508 bore tenor trombones.

That advice makes more sense according to my understanding of the physics -- the entire partial might be slightly off and every position in the partial should be adjusted similarly.

The just/equal temperament thing lead me to suspect that it was the 5th partial (a major 3rd partial, the D) that would be the one most likely to be off, but a trombone is neither a perfect cylinder nor a perfect cone so simplified models might be off. The perfect 5th (aka both the F partials) is pretty exact on an ideal model, but a real trombone isn't ideal.